A sequence of quasipolynomials arising from random numerical semigroups

A numerical semigroup is a subset of the non-negative integers that is closed under addition. For a randomly generated numerical semigroup, the expected number of minimum generators can be expressed in terms of a doubly-indexed sequence of integers, denoted $h_{n, i}$, that count generating sets with certain properties. We prove a recurrence that implies the sequence $h_{n,i}$ is eventually quasipolynomial when the second parameter is fixed.